How do I model this chrysanthemum crest?
It looks like while I was composing my answer someone else beat me to the punch. Nonetheless, I will share what I have. I think you get the gist of it already, but there are some slight differences. I hope you'll find it helpful.

Create the edge outline.



Do a spin extrusion.



Make some Loop Cuts.



Scale down with proportional editing enabled. (Sharp falloff)



Simplify the geometry by merging vertices...



...and dissolving edges.



Add a UV Sphere with the appropriate number of divisions.



Delete the bottom verts.



Do Proportional Editing on the UV (hemi)Sphere too. (Spherical falloff)



Delete the lowest edge loop of the hemisphere.



Merge the pedal's vertices to the center hemisphere.





Duplicate and rotate by 22.5° (360° / 16 pedals)



Merge more vertices.





It may be necessary to Remove Doubles from time to time.



Create a tiling section and use the Array Modifier to complete the circle.

An Empty can be used to specify the rotation angle (22.5°).

Enable the modifier settings "Merge" and "First Last" in order to connect the verts of the mesh as one.



Create the geometry for the smaller pedals.



And give them some height.



Here is the final result: